[1]}, by contrast, specify an approximate likelihood for all units in the population on the observed sample as a function of the conditional inclusion probabilities for the convenience sample, which receives a (logistic regression) model. Since they do not specify a relationship between the propensity score and conditional inclusion probabilities they are unable to directly specify an exact likelihood on the observed sample. Instead, they replace the sum of log-likelihood terms over all population units with a sample weighted approximation summed over the reference sample using the inverse of conditional inclusion probabilities into the reference sample as weights. While their estimator would be expected to produce an unbiased estimation of the conditional inclusion probabilities into the convenience sample, their variance would expect to be inflated due to their use of a pseudo likelihood for the observed sample.
